Revista de la Union Matematica Argentina Volumen 29, 1984.

241

SOME PROXIMITY RELATIONS IN A PROBABILISTIC METRIC SPACE(*)

C. Alsina and E. Trilias

O. INTRODUCTION.

Proximities in a probabilistic metric space have been studied pre­viously by R. Fritsche [3], Gh. Constantin and V. Radu [2] and A.Leog te [4]. In this paper we introduce, using some results concerning o£ der and weak convergences [1], a family of semi-proximities {5~; ~ E A+} analyzing when they are Efrernovi~-proximities and rela-

ting the induced closure operators {C 5 ; ~ E 6+} to those of R, Tar-~

diff [8] and B. Schweizer [7]. In the last section we exhibit a uni-form topology where the neighborhood of a point p is precisely the

closure of {p} in the topology generated by C6 . ~

1. PRELIMINARIES.

Let A+ be the set of all one-dimensional positive distribution func­

tions, i.e., let

A+ = {F: R ... [0,1]; FCO) = 0, F is non-decreasing and left-continuous}.

6+ has a partial order, namely, F ~ G iff F(x) ~ G(x), for every x.

(A+,';;;) is a complete lattice with minimum element £",,(x)

every x, and maximum element the step functlon given by

{ ° , for x .;;; ° £o(x) = • 1 , for x > °

0, for

(1. 1 )

It is well known that weak convergence (w-lim Fn) . in A+ is metriza-n-+oo

ble by the modified Levy metric .c introduced by Sibley [6] .

(*) Presented at the INTERNATIONAL CONGRESS OF MATHEMATICIANS, Hel­sinki. Finland 1978.

242

DEFINITION 1.1. A triangle function is a two-place function T from r:,+ x r:,+ into 6+ such that, for all F, G and H in 6+,

i) T(F,E O) = F,

ii) T(F,G) ~ T(F,H) whenever G ~ H,

iii) T(F,T(G,H)) = T(T(F,G),H),

iv) T(F,G) = T(G,F).

A triangle function T is continuous if it is a continuous function from r:,+ x 6+ into r:,+, where 6+ is indowed with the L-metric topology and r:,+ x6+ with theproduct topology. For a complete study of the fun damental topological semigroups (r:,+,T) see [6].

DEFINITION 1.2. A probabilistic metric space (briefly, a PM-space) is an ordered pair (5 ,F), where 5 is a set, and F is a mapping from 5 x 5 into r:,+ such that for all p,q,r E S:

I) F (p, q)

II) F(p,q)

EO iff p=q,

F(q,p) ,

III) T(F(p,q) ,F(q,r)) .,;; F(p,r).

If F satisfies just (I) and (II) we say that (5,F) is a semi-PM space.

The function F(p,q) is denoted by F ,and F (x), for x> 0, is in-pq pq

terpreted as the probability that the distance between p and q is less than x.

We collect some definitions "about proximities which will be used in the sequel. For a complete survey of proximities see [5] .

DEFINITION 1.3. Let X be a set and 0 a binary relation on P(X), the power set of X. 0 is a semi-proximity if satisfies, for A, Band C subsets of X, the following conditions:

1) 0 ~ A,

2) If AnB # 0 then Ao B,

3) A 0 B implies BoA,

4) A"o (B U C), if and only if A /) B or A /) C.

A semi-proximity 0 is called an Efremovi~ proximity if verifies the ad ditional axiom:

5) A ~ B implies there exists E C X such that E "B and (X-E) lA.

A semi-proximity 0 is said to be separated if

6) aob implies a=b.

Any semi-proximity 0 induces a mapping Co from peX) into itself defi­

ned by CoCA) = {x E X; x 0 AL Co satisEes the conditions

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a) CI)(0) = 0,

b) CI)(A) ~ A, for every A E P(I),

c) CI)(AUB) = CI)(A) UCI)(B) for all A,B E P(I),

i.e., CI) is a Cech closure operator which is a Kuratowski closure (CI)(CI)(A)) = CI)(A) for every A E P(I)) whenever I) is an Efremovic proximity. So I) provides a topology on I called the topo~ogy induced

by o. The topological spaces whose topologies can be derived in this way from proximities are called proximizab~e.

Finally, we summarize some definitions and theorems about order and weak convergences (see [1]).

The supremums and infimums of two functions F,G E ~+, in the lattice (~+,.;;) will be denoted, respectively, by F v G and F" G.

DEFINITION 1.4. (a) A non-decreasing (resp., non-increasing) sequence (Gn) in ~+ is order convergent to G E ~+, if and only if

'" G = V Gn (resp., G n=l

A Gn). (b) A sequence (Fn) in ~+ is order n=l

convergent to F E ~+ (F = o-lim Fn), if and only if there exist two n-+-'"

'" sequences (Gn) and (Hn) such that (Gn) is non-decreasing with n~l Gn

= F, (Hn) is non-increasing with A Hn n=l

Gn .;; Fn .;; Hn. The order limit is unique.

F, and for all n E N is

THEOREM 1.1. Let (Fn) be a sequence in ~+ and F E ~+. Then we have:

i) F = o-lim F iff lim Fn(x) = F(x), for a~l x E R+ (pointwise con n+oo n n-+Ol

vergenae);

ii) If F = o-lim Fn'then F = w-lim Fn n+oo n+oo

does not hold in general;

I-lim Fn , but the reaiproaal n-+-'"

iii) If F = W-!!: Fn and F is continuous or (Fn) is non-deareasing

then F = o-lim Fn. n-+-'"

THEOREM 1.2. (Weak version of Everett diagonal condition in ~+). Let

(F~)(n,k) e: NxN be a aoZZeation ofsequenaes in ~+. let (Fn) be a se­

quenae in~+ and F E ~+. If F has at most a finite set of disaonti­

nuities. F = o-lim Fn' and for eaah n E N F = o-lim F~ • then there n+oo ~ n k+oo

exists a striatly increasing sequenae of integers

kl < k2 < ... < kn < ... in N, suah that F - 0 1° Fn - - 1m k. n+oo n

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2. A FAMILY OF PROXIMITIES IN A PM-SPACE.

Let (S,F) be a semi-PM space. For each ~ E ~+ we define a binary re­lation 6~ on peS) in the following way, for A,B E peS),

"A 6~ B iff there exists a sequence ((an,bn))n£N in A x B such that

When A 6~ B we will say that A and B have a ~-proximity.

THEOREM 2. 1. 6~ is a semi-proximity.

The ~ech closure induced by 6~ will be:

THEOREM 2.2. If T is aontinuous. T(~,~) : ~ and ~ is aontinuous in R+, then C6 is a Kuratowski aZosure.

~

Proof. If x E C6 (C 6 (A)) there is (xn) C C6 (A) such that ~ ~ ~

o-lim (~AF ):~. For each n EN, xn E C6 (A), i.e., there exists _- ~n . ~

a sequence (a~)kEN C A such thato-kl~~ (~A F n):~. By Theorem 1.2 ~_ xnak

there exists an increasing sequence of integers (kn)nEN such that

Using the continuity of T and ~, we have o-lim Hn : T(~,~) : ~ , and n+-

by the triangle inequality Hn ~ F nand Hn ~ T(~,~) : ~, we will ob xakn

i.e., x E C6 (A). ~

;so;; ~ which in turn implies o-lim (1/1 A F xan ) n+- k n

The following example shows that the strong hypothesis ~ sumed above, is really necessary.

EXAMPLE 2.1. Consider the PM-space (R+,Elx_yl'*) and ~ cUI . The con 2,1

volution * is continuous [6] and has no idempotents different from EO

and E_. It is easy to see that

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In order to analyse the special case ~ lemma.

EO we recall the following

LEMMA 2.1. Let I be any set of indiaes and let {F i ; i E I} be in A+. The follo~ing statements are equivalent:

ii) For any E E (0,1) there exists i E I such that Fi(E) > 1-E;

iii) For any E E (0,1) there exists i E I suah that £(Fi,EOJ < E;

iv) There is a countable subset J of I suah that i~J Fi = EO'

Then the Efremovi~ proximity 6E can be presented in the following o

ways:

"A 6 B iff V Fab = EO iff for every E,A > 0 there is EO (a,b)EAxB

(a,b) E AxB such that Fab(E) > 1-A"

and

C6 (A) EO

where Nx(E,A) = {y E S; F (E) > 1-A} are the neigborhood of the c1as­xy sica1 E,A-topo1ogy for these spaces, i.e., the E,A-topo1ogy is proxi­mizab1e by 6E .

o

THEOREM 2.3. Under the hypothesis of Theorem 2.2, the topological sp~

ae (S,C6~) is completely reguZar.

In a PM-space (S, F, T) and for a fixed ~ E A+, Schweizer [7] has intro duced the next relation in peS):

"A 1~ B iff there exists (a,b) E A x B such that F ab ;;;. ~", and when

A 1~ B, A and B are said to be indistinguishable (mod.~).

We note that 1~ is a semi-proximity weaker than 6~, in the sense that A 1~ B implies A 6~ B, Le., indistinguishability (mod.~) yields ~-proximity. The reciprocal does not hold, in general.

EXAMPLE 2.2. Consider the PM-space (R+,E,*), where E E!p_q! for pq all p,q E R+. Let k > 0 and ~ = Ek · Take A = [ 0,1 ) and B = (1+k,+oo) ..

Taking for each n E N, a = 1-1/n E A and b = 1+k+1/n E B, we have n n

o -lim (E k II E ! a _ b !). = 0 -1 im E = E k ' n~OO n n n~oo k~

n

Le., A 6E B but A ¥E B because for all (a,b) E A x B we have . k k

E! a-b! < Ek ·

Recently, Tardiff has introduced [8] for ~ E A+ a closure operator de

246

fined by

{x E S; (\I h E (0,11)(3 a a(h) E A) such that Fh ;;;'.p} , xa

being

Fh (t) xa 1

0 , if t.;;;O,

min(Fxa (t+h)+h,1), if

1 , if t > 1/h .

The semi-proximity T.p defined by

t E (0,1 /h] ,

"A T.p B iff C.p (A) n C.p (B) f. 0",

is stronger than r.p because if A r.p B then there is (a,b) E A x B such

h that Fab ;;;..p and for all h > 0, Fab ;;;. Fab ;;;..p , i.e., A T.p B. The re-

ciprocal does not hold, in general.

EXAMPLE 2.3. Consider the PM-space of example 2.1 , and the same .p =Ek,

k > 0. Let A =[0,1). Then Cr (A) = [0,1 + k) c C (A) • But 1+kECE (A) Ek k

because, for any h E

{1+k} TE A k

(0,11,

but

Ek

taking 1-h E

{1+k}.£E A'. k

Finally we remark that for .p = EO' TE o

for any .p and pES: Co ({p}) = Cr ({p}) .p .p

A we have h E1+k- 1+h = h :;:, Ek+h P"

0E is the E,A-proximity and o

= C,n ( {p}) = {q E S; F ;;;'.p} , ,.. pq

and this set is exactly the class of p in the partition of S induced by the equivalence relation of indistinguishability (mod . .p) introdu­ced in [7] •

3~ A PROXIMITY INDUCED BY AN UNIFORMITY.

DEFINITION 3.1. A triangular function T is said to be radical if for

any F E ~+-{EO} there exists G E ~+-{EO} such that F < T(G,G) < EO'

THEOREM 3.1. If T ;;;. * then T is radical.

Proof. We need to show that for any F < EO there is G < EO such that

G*G > F. In effect, if F = Ek for some k > ° then taking G = Ek/4 we

have G*G = Ek / 2 > Ek . If F < Ek for some k > 0 then the same G yields

the same conclusion. So we can suppose that there is an interval (O,k) such that F(x) > 0 for x E (O,k). Let

H(x) j ° , +1fiT2XT,

if x .;;; 0,

if ° < x.

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Obviously F(x) ..;; +v'F(XJ ..;; +IF(2x) for x > 0, and consequently F ..;; H. If F (x) = H (x) i6r all x > 0 then F (x) = IF(XT and F (x)(F (x) -1) = 0, i.e., there would exist k' > 0 such that F(x) = £k" which is a con­tradiction. So F < Hand there is at> 0 such that 0 < F(t) < H(t) < < 1. Let

G(x) = { H, (x) if x..;; t,

if x > t.

G > H and a straightforward computation shows that F ..;; H*H. By the strict isotony of *, F ..;; H*H < G*G < £0.

Let (S,F,T) be a PM-space. For any F E A+~{£O}' let U(F) = {(p,q) E S x S; F > F}. pq

THEOREM 3.2. If T is radical then the collection {U(F); F E A+-{£O}}

is a basis for a diagonal separated uniformity U on S.

Proof. Obviously AS = {(p,p); pES} C U(F) and U(F) = U(F)-l, for

any F < £0. If F,G < £0 and being T radical there is G < £0 such that F < T(G,G) < £0. Then U(G) 0 U(G) C U(F) because if (p,r) E U(G) 0 U(G), there is q E S such that (p,q) E"U(G) and (q,r) E U(G). By the trian­gle inequality F ~ T(F ,F ) ~ T(G,G) > F, so (p,q) E U(F). Finally

pr pq qr

note that U is separated because n U(F) AS. F£A+-{£O}

COROLLARY 3.1. The topology generated by U is metrizable.

Proof. Consider the countable family {at,t'; t,t' E (0,1) n Q} C A+,

where

at,t'(x) \

0, if x";;O,

t' if 0 < x ..;; t,

1 if x > t.

If U E U, there is F < £0 such that U(F) cU. Being F < £0 there exists

t E (0,1) n Q suth that F(t) < 1. Let t' E (F(t),1). Then F < at,t'

and U(F) ~ U(at,t')' i.e., {U(at,t'); t,t' E (0,1) n Q} is a.countable

basis for U. We apply then Weyl theorem.

The topology generated by U can be described by the family N(U)

= {Np(F); F E A+~{£O}' PES}, where each neighborhood Np(F) is given

by

i.e., Np(F) is precisely the closure of {p} by C1 ,C& or CF, in F F

other words, if q E N (F) then q is indistinguishable (mod.F) of p. . p

The uniformity U induces a proximity &U defined on peS) by:

"A ¥u B iff for some F < £0' Np(F) n B = 0, for all pEA".

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Applying a well known result of proximity theory we obtain that the topology induced"by aU is the uniform topology. We remark that this topology is exactly the topology TF obtained when considering the PM­space as generalized metric space [9].

REFERENCES

[1] C.ALSINA and E.TRILLAS, O~de~ and wea~ eonve~genee~ 06 d~ht~~bu­t~on 6unet~onh, (to appear).

[2] Gh.CONSTANTIN and V.RADU, Random-p~ox~mal hpaeeh, Pub. Univer. Din Timi~oara (1974).

[3] R.FRITSCHE, P~ox~m~t~eh ~n a p~obab~l~ht~e met~~e hpaee, (unpub.).

[4] A.LEONTE, P~ox~m~tate htoehaht~ea, Stud. Cere. Mat.,26 (1974) 1095-1100.

[5] S.A.NAIMPALLY and B.D.WARRAK, P~ox~m~ty hpaeeh, Cambridge Univ. Press (1970).

[6] B.SCHWEIZER, Mult~pl~eat~onh on the hpaee 06 p~obab~l~ty d~ht~~bu­t~on 6unet~onh, Aeq. Math., 12 (1975), 151-183.

[7] B.SCHWEIZER, Su~ la pOhh~b~l~t~ de d~ht~ngue~ leh po~nth danh un ehpace m~t~~qlle. al~ato~~e., C.R. Acad. Sc.Paris, 280 (1975) 459-461.

[8] R.TARDIFF, Topolog~e.h 60~ p~obab~l~ht~c me.t~~c hpace.h, Pacific Jour. of Math., 65 (1976), 233-251.

[9] E.TRILLAS and C.ALSINA, Int~oduee~6n a 10h e.hpae~oh m~t~~coh ge.ne.­~al~zadoh, Serie Univ., n049 (Fundacion Juan March). Madrid 1978.

Recibido en diciembre de1979.

Department of Mathematics and Statistics E.T.S. Arquitectura. Universitat Politecnica de Barcelona. Avenida Diagonal 649 Barcelona, Spain.